from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1723, base_ring=CyclotomicField(1722))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1723))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1723\) | |
| Conductor: | \(1723\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1722\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{861})$ |
| Fixed field: | Number field defined by a degree 1722 polynomial (not computed) |
First 31 of 480 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1723}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{574}\right)\) | \(e\left(\frac{1}{1722}\right)\) | \(e\left(\frac{69}{287}\right)\) | \(e\left(\frac{415}{574}\right)\) | \(e\left(\frac{104}{861}\right)\) | \(e\left(\frac{409}{574}\right)\) | \(e\left(\frac{207}{574}\right)\) | \(e\left(\frac{1}{861}\right)\) | \(e\left(\frac{242}{287}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1723}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{509}{574}\right)\) | \(e\left(\frac{415}{1722}\right)\) | \(e\left(\frac{222}{287}\right)\) | \(e\left(\frac{25}{574}\right)\) | \(e\left(\frac{110}{861}\right)\) | \(e\left(\frac{405}{574}\right)\) | \(e\left(\frac{379}{574}\right)\) | \(e\left(\frac{415}{861}\right)\) | \(e\left(\frac{267}{287}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1723}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{491}{574}\right)\) | \(e\left(\frac{839}{1722}\right)\) | \(e\left(\frac{204}{287}\right)\) | \(e\left(\frac{341}{574}\right)\) | \(e\left(\frac{295}{861}\right)\) | \(e\left(\frac{473}{574}\right)\) | \(e\left(\frac{325}{574}\right)\) | \(e\left(\frac{839}{861}\right)\) | \(e\left(\frac{129}{287}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{1723}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{574}\right)\) | \(e\left(\frac{209}{1722}\right)\) | \(e\left(\frac{71}{287}\right)\) | \(e\left(\frac{61}{574}\right)\) | \(e\left(\frac{211}{861}\right)\) | \(e\left(\frac{529}{574}\right)\) | \(e\left(\frac{213}{574}\right)\) | \(e\left(\frac{209}{861}\right)\) | \(e\left(\frac{66}{287}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{1723}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{177}{574}\right)\) | \(e\left(\frac{901}{1722}\right)\) | \(e\left(\frac{177}{287}\right)\) | \(e\left(\frac{241}{574}\right)\) | \(e\left(\frac{716}{861}\right)\) | \(e\left(\frac{1}{574}\right)\) | \(e\left(\frac{531}{574}\right)\) | \(e\left(\frac{40}{861}\right)\) | \(e\left(\frac{209}{287}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{1723}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{381}{574}\right)\) | \(e\left(\frac{1453}{1722}\right)\) | \(e\left(\frac{94}{287}\right)\) | \(e\left(\frac{295}{574}\right)\) | \(e\left(\frac{437}{861}\right)\) | \(e\left(\frac{187}{574}\right)\) | \(e\left(\frac{569}{574}\right)\) | \(e\left(\frac{592}{861}\right)\) | \(e\left(\frac{51}{287}\right)\) | \(e\left(\frac{8}{21}\right)\) |
| \(\chi_{1723}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{571}{574}\right)\) | \(e\left(\frac{1123}{1722}\right)\) | \(e\left(\frac{284}{287}\right)\) | \(e\left(\frac{531}{574}\right)\) | \(e\left(\frac{557}{861}\right)\) | \(e\left(\frac{107}{574}\right)\) | \(e\left(\frac{565}{574}\right)\) | \(e\left(\frac{262}{861}\right)\) | \(e\left(\frac{264}{287}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1723}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{517}{574}\right)\) | \(e\left(\frac{1247}{1722}\right)\) | \(e\left(\frac{230}{287}\right)\) | \(e\left(\frac{331}{574}\right)\) | \(e\left(\frac{538}{861}\right)\) | \(e\left(\frac{311}{574}\right)\) | \(e\left(\frac{403}{574}\right)\) | \(e\left(\frac{386}{861}\right)\) | \(e\left(\frac{137}{287}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1723}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{574}\right)\) | \(e\left(\frac{1367}{1722}\right)\) | \(e\left(\frac{187}{287}\right)\) | \(e\left(\frac{193}{574}\right)\) | \(e\left(\frac{103}{861}\right)\) | \(e\left(\frac{27}{574}\right)\) | \(e\left(\frac{561}{574}\right)\) | \(e\left(\frac{506}{861}\right)\) | \(e\left(\frac{190}{287}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{1723}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{375}{574}\right)\) | \(e\left(\frac{829}{1722}\right)\) | \(e\left(\frac{88}{287}\right)\) | \(e\left(\frac{209}{574}\right)\) | \(e\left(\frac{116}{861}\right)\) | \(e\left(\frac{401}{574}\right)\) | \(e\left(\frac{551}{574}\right)\) | \(e\left(\frac{829}{861}\right)\) | \(e\left(\frac{5}{287}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1723}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{215}{574}\right)\) | \(e\left(\frac{1409}{1722}\right)\) | \(e\left(\frac{215}{287}\right)\) | \(e\left(\frac{403}{574}\right)\) | \(e\left(\frac{166}{861}\right)\) | \(e\left(\frac{559}{574}\right)\) | \(e\left(\frac{71}{574}\right)\) | \(e\left(\frac{548}{861}\right)\) | \(e\left(\frac{22}{287}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{1723}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{423}{574}\right)\) | \(e\left(\frac{1229}{1722}\right)\) | \(e\left(\frac{136}{287}\right)\) | \(e\left(\frac{323}{574}\right)\) | \(e\left(\frac{388}{861}\right)\) | \(e\left(\frac{411}{574}\right)\) | \(e\left(\frac{121}{574}\right)\) | \(e\left(\frac{368}{861}\right)\) | \(e\left(\frac{86}{287}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1723}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{387}{574}\right)\) | \(e\left(\frac{929}{1722}\right)\) | \(e\left(\frac{100}{287}\right)\) | \(e\left(\frac{381}{574}\right)\) | \(e\left(\frac{184}{861}\right)\) | \(e\left(\frac{547}{574}\right)\) | \(e\left(\frac{13}{574}\right)\) | \(e\left(\frac{68}{861}\right)\) | \(e\left(\frac{97}{287}\right)\) | \(e\left(\frac{10}{21}\right)\) |
| \(\chi_{1723}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{574}\right)\) | \(e\left(\frac{769}{1722}\right)\) | \(e\left(\frac{253}{287}\right)\) | \(e\left(\frac{565}{574}\right)\) | \(e\left(\frac{764}{861}\right)\) | \(e\left(\frac{543}{574}\right)\) | \(e\left(\frac{185}{574}\right)\) | \(e\left(\frac{769}{861}\right)\) | \(e\left(\frac{122}{287}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{1723}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{574}\right)\) | \(e\left(\frac{1391}{1722}\right)\) | \(e\left(\frac{121}{287}\right)\) | \(e\left(\frac{395}{574}\right)\) | \(e\left(\frac{16}{861}\right)\) | \(e\left(\frac{85}{574}\right)\) | \(e\left(\frac{363}{574}\right)\) | \(e\left(\frac{530}{861}\right)\) | \(e\left(\frac{258}{287}\right)\) | \(e\left(\frac{10}{21}\right)\) |
| \(\chi_{1723}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{574}\right)\) | \(e\left(\frac{1717}{1722}\right)\) | \(e\left(\frac{229}{287}\right)\) | \(e\left(\frac{221}{574}\right)\) | \(e\left(\frac{341}{861}\right)\) | \(e\left(\frac{251}{574}\right)\) | \(e\left(\frac{113}{574}\right)\) | \(e\left(\frac{856}{861}\right)\) | \(e\left(\frac{225}{287}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1723}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{507}{574}\right)\) | \(e\left(\frac{1355}{1722}\right)\) | \(e\left(\frac{220}{287}\right)\) | \(e\left(\frac{379}{574}\right)\) | \(e\left(\frac{577}{861}\right)\) | \(e\left(\frac{285}{574}\right)\) | \(e\left(\frac{373}{574}\right)\) | \(e\left(\frac{494}{861}\right)\) | \(e\left(\frac{156}{287}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1723}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{465}{574}\right)\) | \(e\left(\frac{431}{1722}\right)\) | \(e\left(\frac{178}{287}\right)\) | \(e\left(\frac{351}{574}\right)\) | \(e\left(\frac{52}{861}\right)\) | \(e\left(\frac{61}{574}\right)\) | \(e\left(\frac{247}{574}\right)\) | \(e\left(\frac{431}{861}\right)\) | \(e\left(\frac{121}{287}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1723}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{574}\right)\) | \(e\left(\frac{925}{1722}\right)\) | \(e\left(\frac{111}{287}\right)\) | \(e\left(\frac{443}{574}\right)\) | \(e\left(\frac{629}{861}\right)\) | \(e\left(\frac{59}{574}\right)\) | \(e\left(\frac{333}{574}\right)\) | \(e\left(\frac{64}{861}\right)\) | \(e\left(\frac{277}{287}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1723}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{209}{574}\right)\) | \(e\left(\frac{785}{1722}\right)\) | \(e\left(\frac{209}{287}\right)\) | \(e\left(\frac{317}{574}\right)\) | \(e\left(\frac{706}{861}\right)\) | \(e\left(\frac{199}{574}\right)\) | \(e\left(\frac{53}{574}\right)\) | \(e\left(\frac{785}{861}\right)\) | \(e\left(\frac{263}{287}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1723}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{574}\right)\) | \(e\left(\frac{313}{1722}\right)\) | \(e\left(\frac{72}{287}\right)\) | \(e\left(\frac{171}{574}\right)\) | \(e\left(\frac{695}{861}\right)\) | \(e\left(\frac{15}{574}\right)\) | \(e\left(\frac{503}{574}\right)\) | \(e\left(\frac{313}{861}\right)\) | \(e\left(\frac{265}{287}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{1723}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{443}{574}\right)\) | \(e\left(\frac{439}{1722}\right)\) | \(e\left(\frac{156}{287}\right)\) | \(e\left(\frac{227}{574}\right)\) | \(e\left(\frac{23}{861}\right)\) | \(e\left(\frac{463}{574}\right)\) | \(e\left(\frac{181}{574}\right)\) | \(e\left(\frac{439}{861}\right)\) | \(e\left(\frac{48}{287}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{1723}(105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{159}{574}\right)\) | \(e\left(\frac{751}{1722}\right)\) | \(e\left(\frac{159}{287}\right)\) | \(e\left(\frac{557}{574}\right)\) | \(e\left(\frac{614}{861}\right)\) | \(e\left(\frac{69}{574}\right)\) | \(e\left(\frac{477}{574}\right)\) | \(e\left(\frac{751}{861}\right)\) | \(e\left(\frac{71}{287}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1723}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{391}{574}\right)\) | \(e\left(\frac{197}{1722}\right)\) | \(e\left(\frac{104}{287}\right)\) | \(e\left(\frac{247}{574}\right)\) | \(e\left(\frac{685}{861}\right)\) | \(e\left(\frac{213}{574}\right)\) | \(e\left(\frac{25}{574}\right)\) | \(e\left(\frac{197}{861}\right)\) | \(e\left(\frac{32}{287}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1723}(115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{574}\right)\) | \(e\left(\frac{683}{1722}\right)\) | \(e\left(\frac{59}{287}\right)\) | \(e\left(\frac{463}{574}\right)\) | \(e\left(\frac{430}{861}\right)\) | \(e\left(\frac{383}{574}\right)\) | \(e\left(\frac{177}{574}\right)\) | \(e\left(\frac{683}{861}\right)\) | \(e\left(\frac{261}{287}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1723}(116,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{574}\right)\) | \(e\left(\frac{1315}{1722}\right)\) | \(e\left(\frac{43}{287}\right)\) | \(e\left(\frac{425}{574}\right)\) | \(e\left(\frac{722}{861}\right)\) | \(e\left(\frac{571}{574}\right)\) | \(e\left(\frac{129}{574}\right)\) | \(e\left(\frac{454}{861}\right)\) | \(e\left(\frac{234}{287}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{1723}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{365}{574}\right)\) | \(e\left(\frac{1511}{1722}\right)\) | \(e\left(\frac{78}{287}\right)\) | \(e\left(\frac{257}{574}\right)\) | \(e\left(\frac{442}{861}\right)\) | \(e\left(\frac{375}{574}\right)\) | \(e\left(\frac{521}{574}\right)\) | \(e\left(\frac{650}{861}\right)\) | \(e\left(\frac{24}{287}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{1723}(120,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{574}\right)\) | \(e\left(\frac{145}{1722}\right)\) | \(e\left(\frac{247}{287}\right)\) | \(e\left(\frac{479}{574}\right)\) | \(e\left(\frac{443}{861}\right)\) | \(e\left(\frac{183}{574}\right)\) | \(e\left(\frac{167}{574}\right)\) | \(e\left(\frac{145}{861}\right)\) | \(e\left(\frac{76}{287}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{1723}(122,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{517}{574}\right)\) | \(e\left(\frac{673}{1722}\right)\) | \(e\left(\frac{230}{287}\right)\) | \(e\left(\frac{331}{574}\right)\) | \(e\left(\frac{251}{861}\right)\) | \(e\left(\frac{311}{574}\right)\) | \(e\left(\frac{403}{574}\right)\) | \(e\left(\frac{673}{861}\right)\) | \(e\left(\frac{137}{287}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1723}(127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{571}{574}\right)\) | \(e\left(\frac{1697}{1722}\right)\) | \(e\left(\frac{284}{287}\right)\) | \(e\left(\frac{531}{574}\right)\) | \(e\left(\frac{844}{861}\right)\) | \(e\left(\frac{107}{574}\right)\) | \(e\left(\frac{565}{574}\right)\) | \(e\left(\frac{836}{861}\right)\) | \(e\left(\frac{264}{287}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{1723}(129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{574}\right)\) | \(e\left(\frac{719}{1722}\right)\) | \(e\left(\frac{247}{287}\right)\) | \(e\left(\frac{479}{574}\right)\) | \(e\left(\frac{730}{861}\right)\) | \(e\left(\frac{183}{574}\right)\) | \(e\left(\frac{167}{574}\right)\) | \(e\left(\frac{719}{861}\right)\) | \(e\left(\frac{76}{287}\right)\) | \(e\left(\frac{10}{21}\right)\) |